AI 中文总结
该研究提出球上平方和松弛的Argmax原理，统一多项SoS收敛分析，针对最佳可分态、矩阵$2\to4$范数等问题给出更优或更简洁的SoS分析结果。
AI 中文摘要
我们开发了一种Argmax原理，用于分析单位球上优化问题的平方和（Sum-of-Squares, SoS）松弛。给定一个可行的伪期望，我们构造高阶伪矩多项式，例如$\u03a6_k(u)=\widetilde{\mathbb E}\langle x,u\rangle^{2k}$。我们的指导原则是，其最大化者是取整候选者：它们的局部和全局最优性条件揭示了支配SoS收敛的重加权伪期望不等式。这一观点统一了此前用截然不同技术分析的几个问题。我们得到三个结果：第一，针对最佳可分态（Best Separable State），我们在完美完备 regime 下给出了用于近似$h_{\mathrm{sep}}(P)$的$O(\sqrt{n/\epsilon})$阶SoS分析，改进并简化了Barak、Kothari和Steurer（STOC'17）的工作；在指数时间假设（Exponential-Time Hypothesis）下，逆线性间隙的依赖性本质上是紧的，与QMA(2)协议的困难性匹配。第二，针对矩阵$2\to4$范数，$O(\sqrt{n}/\epsilon)$阶SoS给出了$(1+\epsilon)$的乘性近似；Barak等人（STOC'12）此前给出了可比时间的常数间隙决策算法，我们的结果提供了乘性保证，并扩展到$q$为偶数的$p\to q$范数族。最后，针对$d$阶多项式优化，我们用更简短、更直接的证明重现了Bhattiprolu等人（FOCS'17）的收敛定理：$k$阶SoS给出$O_d((n/k)^{d/2-1})$的近似比。本文未引入新的松弛，而是通过高阶矩Argmax提供了读取SoS解的通用方式，统一了此前独立的收敛分析，并得到更精确的界或更简洁的证明。
英文摘要
We develop an argmax principle for analyzing sum-of-squares relaxations of optimization problems over the unit sphere. Given a feasible pseudo-expectation, we form a polynomial of high-order pseudo-moments, such as $Φ_k(u)=\widetilde{\mathbb E}\langle x,u\rangle^{2k}$. Our guiding principle is that its maximizers are rounding candidates: their local and global optimality conditions reveal the reweighed pseudo-expectation inequalities governing SoS convergence. This viewpoint unifies several problems previously analyzed by rather different techniques. We obtain three results. First, for Best Separable State, we give a degree-$O(\sqrt{n/ε})$ SoS analysis for approximating $h_{\mathrm{sep}}(P)$ in the perfect-completeness regime, improving and simplifying Barak, Kothari and Steurer (STOC'17). The dependence is essentially tight for inverse-linear gap under the Exponential-Time Hypothesis, matching hardness from $\mathrm{QMA}(2)$ protocols. Second, for the matrix $2\to4$ norm, degree-$O(\sqrt n/ε)$ SoS gives a multiplicative $(1+ε)$ approximation. Barak et al. (STOC'12) previously gave a comparable-time constant-gap decision algorithm; our result gives a multiplicative guarantee and extends to a family of $p\to q$ norms with even $q$. Finally, for degree-$d$ polynomial optimization, we recover the convergence theorem of Bhattiprolu et al. (FOCS'17) with a shorter, more direct proof: degree-$k$ SoS gives approximation ratio $O_d((n/k)^{d/2-1})$. The paper introduces no new relaxation. Instead, the high-moment argmax gives a common way to read an SoS solution, unifying previously separate convergence analyses and yielding sharper bounds or simpler proofs.
CommentsSubmitted to SODA
